Mechanical Properties

Creep Mechanisms

1) Dislocation Creep 2) Nabrro-Herring Creep 3) Coble Creep Both Nabarro-Herring and Coble creep occur because under the application of stress, grain boundaries normal to the applied stress experience tensile stresses and thus develop a higher concentration of vacancies. Grain boundaries parallel to the applied stress will experience compressive stresses and will thus have lower concentration vacancy. This gradient in vacancy concentration leads to a flux of vacancies diffusing from the perpendicular grain boundaries to the parallel grain boundaries. It requires higher temperatures to activate Nabaroo-Herring creep, since it requires more energy to transport vacancies through grains than along grain boundaries.

Brittle

A brittle material has very low ductility, meaning it tends to fracture at low strains after little plastic deformation. Brittle materials are considered to be those with a fracture strain of <~5%.

Compression Testing

A compression test is another type of mechanical testing in which a uniaxial compressive stress is applied to a specimen held between two plates. For most metals, the compressive strength will be very similar to the tensile strength. However, brittle materials, including ceramics, brittle metals, concrete, etc. can be more easily tested with compression, and will have much higher compressive strengths than tensile strengths. Similar to in tensile testing, the cross sectional area of the specimen will change during loading, however in this case the specimen will shorten and its area will become larger. Since there is on clear fracture, compression testing is not a useful method for determining the fracture point of a material

Creep Test

A creep test is generally done under uniaxial tensile stress. A creep curve is plotted as creep strain vs. time at a constant load and temperature.

Texture

A polycrystalline material is composed of many grains that can be oriented in any direction. Texture refers to the distribution of crystallographic orientations in a polycrystalline material.

Ashby Map

An Ashby map is a plot of applied stress versus temperature which shows the conditions under which different creep mechanisms dominate.

Fatigue Curve

At any given amplitude of applied stress, it will require a different number of cycles to result in failure. If the stress amplitude (S) is plotted against the number of cycles to failure (N), the result is an S-N curve: The S-N curve indicates that the higher the magnitude of the stress, the smaller the number of cycles that material is capable of sustaining before failure, and vice versa. For some ferrous materials and titanium alloys, the S-N curve becomes horizontal at higher N values, or there is a limiting stress level called the "fatigue limit" or "endurance limit", which is the largest value of fluctuating stress that will not cause failure for essentially an infinite number of cycles. For many steels, fatigue limits range between 35% and 60% of the tensile strength. However, most other alloys show a gradually sloping S-N curve.

Dislocation Creep

At high stresses, creep is controlled by dislocation motion. This is accomplished by both dislocation glide and climb, however climb is the rate-controlling element in creep. The glide motion of dislocations is impeded by long range stresses due to dislocation interactions. These stresses can be relieved by dislocation climb and subsequent annihilation.

Stresses in thin walled tube

Azimuthal force: 𝜎𝜃 = 𝑃𝑅/ δ axial force: 𝜎𝑍 = 𝑃𝑅/2δ P=internal gas pressure R=tube radius δ=tube thickness. The azimuthal force, 𝜎𝜃, acts around the circumference of the tube, and the axial force, 𝜎𝑍 acts along the tube length. It is clear that 𝜎𝜃 = 2𝜎𝑍, so the azimuthal force is always a factor of 2 larger than the axial force. This explains why tubes are observed to rupture along the length axis with a rupture in the circumference (like sausages on a grill!).

Plastic Regime (nonuniform strain)

Beyond the UTS, plastic deformation is non-uniform, also called necking. Unlike a uniform reduction in cross sectional area, a specimen undergoing necking "necks down" and becomes thin at a specific location

Brittle failure

Brittle failure is characterized by rapid rate of crack propagation (unstable) with no gross deformation.

Coble Creep

Creep due to the diffusion of vacancies along grain boundaries.

Nabarro-Herring Creep

Creep due to the diffusion of vacancies through the lattice (through grains)

Solid Solution Strengthening

Dislocations encounter barrier on their path from the solutes present in the host lattice

Precipitation hardening

Dispersed particles impart significant obstacles on dislocation motion. When the particles are small (<5 nm), and/or soft and coherent, dislocations can cut and deform the particles ("particle shearing"). For larger precipitates, the phenomenon of Orowan bypassing occurs, in which the dislocations form loops around the particles.

Types of failure

Ductile Brittle

Ductile Failure

Ductile fracture is characterized by appreciable plastic deformation with stable crack growth. A tell-tale sign of ductile fracture in tensile testing is the coneshaped surface of a specimen shown above.

Ductility

Ductility is a measure of the degree of plastic deformation that can be sustained at fracture. A metal that withstand a large amount of plastic deformation before fracturing is called a ductile material. The ductility of a material based on tensile testing can be expressed quantitatively as a percent elongation of the plastic strain at fracture.

Knoop (microhardness)

Due to the unique shape of the Knoop indenter, elastic recover of the projected impression occurs in a transverse direction, that is, along the shorter diagonal length, rather than the long diagonal. Therefore, the measured longer diagonal length will give a hardness value close to what is given by the uncovered impression. The Knoop hardness number (KHN) is given by the load divided by the unrecovered projected are of the impression (not the surface area of the indentation): 𝐾𝐻𝑁 = 𝑃/𝐴p = 𝑃/𝐶𝑙^2 where 𝑙 is the longer diagonal length (mm) and C is the Knoop indenter constant that relates the longer diagonal length to the unrecovered projected area (generally 0.07028).

Types of Deformation

Elastic Plastic

Stress-Strain curve regions

Elastic Regime Yield Point Plastic Regime (uniform strain) Plastic Regime (nonuniform strain)

Elastic Theory

Elasticity theory concerns stresses below the yield stress of a material, or strictly deformation in the elastic regime. One can use it to solve problems by identifying either the applied stresses or the strains occurring in a body, and then using a number of conditions to solve for relations and values of unknown variables. There are 3 main equations and conditions that can be used: (1) equilibrium conditions - relate the stress components to ensure that there is balance of forces on a volume element of the body (2) relations between displacements and strains which ensure the body remains continuous as it deforms (3) constitutive equations: relate stresses and strains based on the elastic properties of the material, such as the Young's modulus, shear modulus, and Poisson's ratio.

Von Mises stress model

Elasticity theory quickly becomes very complex, and there a myriad of full texts written on the subject of applying the process to different scenarios. Typically, loading conditions are NOT uniaxial, making defining the stress components complex. Therefore it is easier to devise a method to determine which combination of stress components is equivalent to a uniaxial stress. This is what the von Mises model was developed for

Microhardness

For very small objects, thin films, and surface hardened materials, it is useful to decrease the indent size and depth. There are two standard microhardness tests, Vickers and Knoop.

stress

Force per unit area. Typically measured in MPa. 𝜎 = 𝐹 /A

Grain size strengthening

Grain boundaries can act as obstacles to dislocation motion. As grain orientation changes at the grain boundary, slip planes in a grain get disrupted at the grain boundary. Thus the dislocations gliding on a slip plane cannot burst through the grain boundary. Instead, they get piled up against it. A greater stress concentration is created at a larger grain, which has enough energy to push through to the next grain. For smaller grains, the stress concentration is not great enough to produce slip in the next grain readily, thus fine grained materials exhibit higher yield strength. 𝜎𝑌𝑆 = 𝜎𝑖 +(𝑘𝛾 /√d) where 𝜎𝑌𝑆 is the yield stress, 𝜎𝑖 is the friction stress (or yield strength at infinite grain size), 𝑘𝛾 is the unlocking parameter (that measures relative hardening contribution of the grain boundaries, and 𝑑 is the grain diameter.

Hardness Testing

Hardness testing is a simple and useful technique to characterize mechanical properties in materials, particularly the resistance to deformation. It does not involve total destruction of the sample as needed in tensile testing, and generally requires only a small volume of material. Hardness is not the same as strength, however hardness values are generally proportional to the strength values obtained in tension or compression tests. There are many types of hardness testing. Macrohardness testing methods include the Brinell hardness test, the Rockwell test, and Macro-Vickers. Microhardness tests are generally done with Vickers indentation and Knoop indentation.

Impact properties

Impact tests are used to measure the resistance to failure of a material under a sudden applied force. They are intended to study the most severe conditions relative to the potential of fracture: (i) deformation at low temperatures (ii) high deformation rate (iii) a triaxial state of stress (accomplished by introducing a notch in the specimen. Impact tests measure the impact energy, or the energy absorbed prior to failure).

anistropy

In a material with no texture, the random orientations of all of the grains average out to result in a net isotropic material. However, strong texture in a material can result in anisotropy, or directional dependence, of mechanical properties. For instance, the material may exhibit higher strength in one direction than in another due to texture. This is particularly evident in materials that have been subject to rolling, which is accomplished by passing metal stock through one or more pairs of rolls to reduce its thickness and make it more uniform. Rolling of metals induces texture and also elongates grains along the rolling direction.

Fatigue Tests

In strain controlled fatigue tests for life evaluation, the cyclic stress strain curve leads to a hysterics loop:

Elastic Regime

In the initial linear regime of the stress-strain curve, the material is experiencing only elastic deformation only. The slope of the linear portion is equal to the Young's modulus of the material.

Upper and Lower Yield points

It is also common for materials to exhibit upper and lower yield points, as shown below. This occurs when dislocation movement gets impeded by interstitial atoms such as carbon, nitrogen, and so on forming solute atmospheres around the dislocations. However, at very high stress, the dislocations break away from the solutes and thus require less stress to move at this point. The elongation that occurs at constant load or stress at the lower yield point is called "yield point elongation". During yield point elongation, a type of deformation bands known as "Lüders bands" are formed across this regime. After the Lüders bands cover the whole gauge length of the specimen, the usual strain hardening regime sets in.

ductile-to-brittle transition temperature (DBTT)

Many materials exhibit both ductile and brittle behavior depending on temperature. For most there is not a single temperature, but rather a range of temperatures over which the ductile-brittle transition occurs. Many materials exhibit both ductile and brittle behavior depending on temperature. For most there is not a single temperature, but rather a range of temperatures over which the ductile-brittle transition occurs. Note: Materials with FCC crystal structure DO NOT exhibit a DBTT. BCC and HCP alloys mainly experience this transition, which is mainly due to highly temperature-sensitive yield stress at lower temperatures. The transition temperature for these materials can depend on alloy composition and microstructure as well. For instance, decreasing the average grain size of a material decreases its DBTT.

3 cracking modes

Mode 1: the displacement is perpendicular to the crack faces Mode 2: the displacement is made parallel to the crack faces, but perpendicular to the leading edge Mode 3: the displacement is parallel to the crack faces and to the leading edge In reality, Mode I (opening mode) is the most important. Conventional tests for fracture toughness are done under Mode I.

Plastic Regime (uniform strain)

Permanent plastic deformation begins to occur in this regime, which is accomplished through the slipping of atomic planes against each other facilitated with dislocation motion. This regime is also referred to as the "strain hardening regime". In this regime, the cross sectional area of the specimen decreases in a uniform manner. The stress continues to increase with strain until it reaches a maximum at the UTS ("ultimate tensile strength") and beyond this point, non-uniform plastic deformation occurs (also called "necking"). The UTS is used often in practice since it is easy to determine, however it is not a property of fundamental significance like the YS.

Strengthening (hardening) Mechanism

Strength is a property which is very sensitive to the microstructure of a material. Dislocations are a common factor in almost all important strengthening mechanisms. If dislocations are able to move in a crystal with relative ease, it means the material does not intrinsically offer any resistance to the dislocation movement and thus would be less strong. But if the microstructure is laden with various obstacles, dislocation movement will be effectively impeded and this movement obstruction will be translated into an increase in strength

Tensile Testing

Tensile (tension) testing is a popular method of studying short-term mechanical behavior (strength and ductility) of a material. The test is performed with a quasi-static uniaxial tensile stress state (the sample is being stretched along one direction at a sufficiently slow and usually constant rate - aka, no sudden changes) Tensile specimens are typically made with standard dimensions according to ASTM guidelines. The raw data from the test is in terms of load (applied force) and elongation (the length the tensile bar was stretched by), and must be converted into stress and strain values.

measuring texture

The amount of texture a material has is determined by measuring crystallographic orientation in the sample over many grains. This can be done with X-ray diffraction, EBSD (electron backscatter diffraction) in an SEM (scanning electron microscope), and other diffraction techniques. These techniques make use of the fact that crystallographic planes of different spacing and orientations with respect to a fixed direction will diffract the incident radiation by an angle related to the crystal orientation.

Thermal Expansion

The azimuthal strain in the tube can be expresses as a component due to the internal pressurizing, as well as a component due to thermal expansion: 𝜀𝜃 = (1/𝐸) (𝜎𝜃 − 𝜈𝜎𝑧 ) + 𝛼(𝑇 − 𝑇𝑟𝑒𝑓) where 𝑇𝑟𝑒𝑓 is a reference temperature and 𝛼 is the thermal expansion coefficient Note: the thermal expansion coefficient is given by: 𝛼 = (Δ𝑥/Δ𝑇) = (𝑢𝑡ℎ − 𝐿𝑜) /(𝑇 − 𝑇𝑟𝑒𝑓) where 𝑢𝑡ℎ is the normal displacement due to thermal expansion, and 𝐿𝑜 is the length at the reference temperature

fracture toughness

The elastic stress field around a crack tip can be described by a single parameter known as the "stress intensity factor", K. It depends on many factors such as the geometry of the crack-containing solids, the size and location of the crack, and the magnitude and distribution of the loads applied. Rapid unstable failure would occur if a critical value of K is reached.

Strain vs Time (Creep Curve)

There are 3 stages in the curve: (1) primary stage (transient creep): work hardening during plastic deformation is more than recovery (softening) exhibiting decreasing strain rate with time (2) secondary or second stage creep (minimum creep rate): the rate of work hardening and softening balance each other (3) tertiary stage: characterized by an accelerating creep rate where softening mechanisms predominate These stages are based on two competing factors: (i) the strain hardening that occurs when something is plastically deformed (due to dislocations getting tangled in each other) (ii) "softening" that occurs as a result of the creep, which causes easy plastic deformation to occur. Each regime is marked by which of these factors wins out.

Load vs Elongation -> true stress vs strain

This calculation of stress vs. strain takes into account the instantaneous values of the cross sectional area and length. Once plastic deformation sets in, the cross sectional area, A, of the specimen begins to decrease. This causes the true stress to continue increasing past necking as the test is continued. Note that although the load does not increase at this point, the true stress continues to due to the decreased cross sectional area that is now being put in tension.

pole figure

To represent the degree and type of texture in a polycrystalline material, the observed grain orientations of the material are all plotted simultaneously in a pole figure. This allows us to view the entire distribution of crystal orientations in one figure.

Poisson's Ratio

When a material is stretched in one direction, it tends to contract in the other two directions (or vice versa for compression). This relationship is captured with Poisson's ratio, 𝜈. If a sample is pulled in tension on the z-direction, Poisson's ratio is given by: 𝜈 = − 𝜀𝑥/𝜀𝑧 = − 𝜀𝑦/𝜀𝑧 where 𝜀𝑧 = strain in the z-direction

strain hardening

When a metal is cold-worked, its strength increases. Generally, an annealed crystal contains a dislocation density of about 108 m−2 . Heavily cold worked materials may contain 1014 to 1016 m−2 . As dislocation density is increased, the movement of dislocations becomes increasingly difficult due to the interfering effect of the stress fields of other dislocations. In the early stages of plastic deformation, slip is generally limited on primary glide planes and the dislocations tend to form coplanar arrays. However, as the deformation proceeds, cross-slip takes place and dislocation multiplication mechanisms start to operate. The cold worked structure then forms high dislocation density regions or tangles that soon develop tangled networks.

Yield Point

When the deformation proceeds past a certain point, it becomes nonlinear. The end of the linear regime is called the "yield point" and the value of the stress at this point is the YS ("yield stress"). The exact value is typically calculated by finding the intersection between the stress-strain curve and a 0.2% offset line of the linear portion of the curve. The yield point marks the transition from elastic to plastic deformation in the material.

Brinell Hardness

a 3000 kg load is applied through a 10 mm diameter hardened steel or tungsten carbide ball. The Brinell Hardness number (BHN) is given by: 𝐵𝐻𝑁 = 2𝑃 /(𝜋𝐷(𝐷 − (√𝐷2 − 𝑑 2))) where P is the applied load in kgf (kg-force), D is the diameter of the ball, and d is the diameter of the impression on the material made by the ball, both in mm.

Vickers hardness

a four-sided pyramid is used that has an angle between the opposite faces of 136°: The Vickers hardness number (VHN), is determined by the load divided by the surface area of the indentation. The area is calculated from the average length of the diagonals, d1 and d2, of the impression. 𝑉𝐻𝑁 = ( 2𝑃 sin 𝜃/2) /( (𝑑1 + 𝑑2)/ 2 )^ 2 = (1.854𝑃)/ ( (𝑑1 + 𝑑2)/ 2 )^2 The units of the VHN are kgf/mm2, and can range from 5-1500. The disadvantages associated with Vickers hardness testing is that it requires good surface preparation. Otherwise there can be errors in the determination of the diagonal length

Rockwell Hardness

a spherical indenter is first used to apply a minor load, followed by a major load. The minor load is used to establish the zero position of the indenter on the sample surface, and then the major load is applied. The major load is then removed while maintaining the minor load. To get the hardness value, the depth of penetration is measured from a dial and used to calculate the hardness. The main advantage of this technique is not having to measure the dimensions of the indentation by hand, but rather having it automatically recorded. There are several different Rockwell scales depending on the type of indenter and major load used. In most cases the hardness is expressed as a dimensionless number.

sheer strain

can be coplanar rather than normal to the sample cross section. 𝛾 = tan 𝜃

shear stress

can be coplanar rather than normal to the sample cross section. 𝜎 = 𝐹 /A

Elastic Deformation

completely recoverable deformation where the body returns to its original shape once the external forces are removed. Atomic bonds are stretched, but not broken.

strong texture

material has a preferred orientation

Bauschinger Effect

refers to the fact that on unloading cycle, the yielding occurs at a lower stress point and in the initial loading (point C compared to point A in the plot above). Continuing this process over many cycles results in the following continuous. In general, the hysteresis loop stabilizes after about 100 cycles, and the stress-strain curve obtained from cyclic loading will be different from that of monotonic loading.

Strain

sample displacement per unit length (%) tension is positive compression is negative 𝜀 = ∆𝐿 /𝐿

Charpy V-notch test

the most common impact test in the U.S. In this test, the load is applied as an impact blow from a weighted pendulum hammer that is released at a fixed position at a fixed height, h1. Upon release, a knife-edge mounted on the pendulum strikes and fractures the specimen at the notch that acts as a stress raiser site for the high-velocity impact blow. After fracturing the specimen, the pendulum continues in its trajectory to reach a height, h2. The energy absorption is calculated from the difference between the pendulum's static energies at h1 and h2: 𝐸 = 𝑀𝑔(ℎ1 − ℎ2) where M is the pendulum's mass

No texture

the orientations are fully random, there is no preferred orientation

Load vs. elongation ->> engineering stress vs. strain

the stress and strain at each point are calculated using only the initial cross sectional area and length of the specimen: 𝜎𝑒 = 𝐹/𝐴0 𝜀𝑒 = 𝐿−𝐿0/𝐿0 This does not take into account the fact that the cross sectional area and length of the specimen are not constant throughout the test.

Creep

time dependent plastic strain at constant temperature and stress Creep occurs at temperatures >0.4-0.5 Tm (the melting point of the material). It is considered to be a thermally activated process, and must be taken into account if a load-bearing structure is exposed to elevated temperatures for a long duration of time.

Fatigue

weakening of a material caused by repeated applied loads (cyclic loading) About 90% of engineering failures are attributed to fatigue of materials under cyclic loads. Fatigue failure occurs after a lengthy period of stress or strain reversals and is brittle-like in nature even in normally ductile materials. Generally, the fracture surface turns out to be perpendicular to the applied stress.

Relationship between Young's Mod (elastic mod) and Sheer Mod

𝐺 = 𝐸/( 2(1 + 𝜈))

Critical Value, K

𝐾𝐼𝑐 = the plain strain fracture toughness The relation for a given type of loading and geometry is: 𝐾𝐼𝑐 = 𝑌𝜎√𝜋𝑎𝑐 Where Y is a parameter that depends on the specimen and crack geometry 𝜎 is the applied stress 𝑎𝑐 is the critical crack length. If 𝐾𝐼𝑐 is known, this can be used to compute the maximum crack length tolerable.

Relationship between stress and strain

𝜎 = 𝐸𝜀 (normal loading) 𝜏 = 𝐺𝛾 (shear loading) E = the Young's modulus, or elastic modulus G = the shear modulus The value of the Young's modulus is determined by the interatomic forces in the material, and is dependent on crystal structure and orientation in the material, however it is not very affected by alloying, cold working, or heat treatment. The Young's modulus does decrease with increasing temperature as the interatomic forces become weaker.